Reprinted from Coastal Dynamics '95: Proceedings of the International Conference on Coastal Research in Terms of Large Scale Experiments, pp.81-9, Gdansk, Poland { September 4-8, 1995 Looking for Wave Groups in the Surf Zone Merrick C. Haller 1 and Robert A. Dalrymple Abstract This paper proposes two new parameters, temporal group steepness (St) and spatial group steepness (Sx), for evaluating wave grouping. The new parameters provide information on two important qualities of groups, the amplitude variation of the groups and their length, both of which have an impact on coastal problems. Computation of the parameters involves the use of the Hilbert transform to determine the wave envelope of incident waves. The Hilbert transform method is reviewed and shown to be superior to other methods involving or jj and the new parameters are shown to be responsive tochanges in the wave eld due to shoaling and wave breaking. The new parameters are applied to eld data to study the variability of wave grouping both spatially, in the cross-shore direction, and temporally, at xed locations. Wave grouping is shown to persist through the surf zone indicating the necessity for the inclusion of nonsteady radiation stress gradients, shoreward of the mean break point, in models of long wave generation. In addition, the temporal change in group steepness is shown to correlate with the increase in long-period surf zone current modulations. 1 Introduction The importance of wave groups is well recognized by the engineering community. Wave groups can have a serious eect on the stability of rubble mound structures (Johnson et al., 1978) and pipelines (Dean, 198). Wave groups inuence the response of ships moored at sea (Hsu and Blenkarn, 197 Pinkster, 1975) and can excite strong harbor oscillations (Bowers, 1988). Wave groups are also 1 Graduate student, Center for Applied Coastal Research, University ofdelaware, Newark, DE 19716, USA, e-mail: merrick@coastal.udel.edu Professor and Director, Center for Applied Coastal Research, University of Delaware, Newark, DE 19716, USA, e-mail: rad@coastal.udel.edu 1
utilized as a forcing mechanism in models for long waves both oshore (Longuet- Higgins and Stewart, 196) and in the nearshore (Tucker, 195 Symonds et al., 198 and others). More recently, laboratory and eld experiments have identied wave groups as a source of edge waves (Bowen and Guza, 1978), rip currents (Tang and Dalrymple, 1989), and possibly shear waves (Hamilton et al., 1995). This paper compares methods of identifying wave groups that are commonly used today and identies some of the problems and inadequacies of existing methods. Amethodofwave group detection involving the Hilbert transform is described and shown to have superior characterisics to competing methods. The method has the advantage of being utilized strictly in the frequency domain. Current parameters used to describe the amount ofwave grouping are evaluated and shown to be of limited usefulness. A new parameter, group steepness, is introduced and evaluated for its usefulness in measuring groupiness in real seas. In contrast to most other studies which have focused on grouping in deep water, this paper will analyze grouping in the surf zone. If the dynamics of the surf zone are expressed byawaveaveraged equation such as @u @t + g @ @x = 1 ; @S xx h @x (1) where u is the cross-shore component ofvelocity, is setup, and S xx is the crossshore component of radiation stress and dependent on the local wave height, then it is evident that wave groups persisting into the surf zone will lead to long wave generation and long-period current modulations. It will be shown that the group steepness in the surf zone has some correlation with observed current oscillations. Identication of Wave Groups In order to quantify the importance of wave groups in the eld a wave group detector must be employed. Wave group detectors take two forms. One form involves counting discrete waves via a zero crossing method. Runs of successive waves higher than an arbitrary threshold wave height are identied and considered a wave group or a \run of waves". In this manner wave groups are treated as isolated events. The second general form of wave group detector involves wave envelope theory. Various techniques are used to determine a continuous measure of wave height (H) orofwave energy (E H ). In this way wave groups are treated as oscillations about a mean value and they can be analyzed in the frequency domain. This is the form of wave group detection that will be utilized herein.
.1 SIWEH One of the well known wave group detectors is the Smoothed Instantaneous Wave Energy History (SIWEH) asdenedby Funke and Mansard (1979). The SIWEH is a running time series of wave energythatisaveraged over the peak wave period by means of a smoothing Bartlett lter. The SIWEH is found by the following: E(t) = 1 Z 1 (t + )Q()d () T p ;1 where E(t) is the energy envelope function, T p is the peak spectral period of the incident waves, and Q() is a Bartlett window smoothing function as dened by: Q() =( 1 ;j j=tp j j <T p j jt p : A similar method, the Local Variance Time Series, was proposed by Thompson and Seelig (1984). It also involves squaring the surface displacement and smoothing over the peak period. There are several diculties inherent to both methods. First, the methods contain no provision for removing low frequency waves from the time series unrelated to the wind/swell record. In the nearshore region, where low frequency waves become highly energetic, the energy time series will be unduly inuenced by such motions. Another diculty arises from the squaring of the surface displacement record, (t). This can be easily shown by rst expressing the wave eld as a Fourier series expansion given as (t) = NX i=1 (3) a i cos(! i t + i ) (4) where! i, a i,and i are the angular frequency, amplitude, and random phase of the individual waves, respectively. Then,asshown by Naess (1978), contains four distinct terms as follows: 1. a mean term proportional to a i. dierence interaction terms with frequencies! i ;! j 3. sum interaction terms with frequencies! i +!j 4. harmonic terms with frequencies! i. It is the rst and second terms which are related to wave grouping. Thus a ltering operation is required to remove the extraneous terms created in the squaring process. In addition, the ltering operation is based on knowledge of the peak period which can be essentially arbitrary in a wide-banded sea. Medina and Hudspeth (1987) demonstrate that the determination of wave group parameters (e.g. GF ) using these methods becomes highly dependent on the ltering operation. In addition, these methods involve performing both envelope detection and ltering in the time domain, which requires heavy CPU time. 3
. Modulus Method Instead of using a time series of wave energy to determine wave group parameters, the envelope signal can be used directly. The wave envelope is essentially a time series of instantaneous wave amplitude. List (199) proposed using the modulus of the measured surface displacement j(t)j to nd the wave envelope. The method can be described as follows: rst, (t) is ltered to remove the lowest frequencies (usually ( :5Hz)) associated with infragravity band waves. Next, the modulus of the resulting time series is determined and multiplied by a normalization factor. The resulting series A(t) j(t)j contains terms 1,, and 4 from the list above. Again, a lowpass lter must be employed on A(t) in order to isolate the dierence interaction terms which are directly related to amplitude modulations of the incident wave eld..3 Hilbert Transform An alternative to both of the previous methods involves the Hilbert transform and can be done quickly and easily in the frequency domain. The use of the Hilbert transform in envelope theory has its origins in the seminal paper on noise in electrical circuits by Rice (1944, 1945). Melville (198) was the rst to apply the method to deep water wave modulation and subsequent work was done by Medina and Hudspeth (1987) among others. The Hilbert transform can be applied in either the time or frequency domains here we perform almost all operations in the frequency domain, utilizing Fast Fourier Transform routines requiring very little CPU time. In the time domain, an analytic signal z(t) can be dened from a given time series (t) and its corresponding Hilbert transform ~(t) z(t) =(t)+i~(t): (5) In the frequency domain the relation between z(t) and(t) is dened in terms of their Fourier transforms as and so, F z (f) = 8 >< >: F for f = F (f) for f> for f< (6) z(t) =F ;1 [F z (f)]: (7) Once the analytic signal is computed we can determine time series of wave energy or wave amplitude from which we can quantify wave grouping. The denition are as follows: A(t) =jz(t)j (8) E(t) =zz = jz(t)j : (9) The most signicant advantage to the Hilbert transform method is that the energy time series contains only two terms: 4
1. a mean term proportional to the variance of (t). dierence interaction terms (! i ;! j ). Thus, the squared envelope function as determined by the Hilbert transform method directly isolates the mean and the dierence interaction terms without ltering. The wave envelope, A(t), gives a direct measure of amplitude modulations that are occuring at all frequencies. Narrow-banded seas will have narrow-banded wave group spectra (S A (f)). The advantage here is that S A (f) only contains amplitude modulations. Hence, for wide-banded seas with widebanded group spectra if ltering is applied to S A (f) only high frequency wave groups are removed (e.g. T 3s) and only low frequency wave groups remain. 3 Quantifying Groupiness 3.1 Run Length Perhaps the most commonly used parameter to quantify wave groupiness is the mean run length, which istheaverage numberof waves per wave group with wave groups being dened as a sequence of waves whose heights exceed a threshold value. Goda (197) related wave group statistics to spectral information in an attempt to achieve predictability. Kimura (198) extended Goda's theory by assuming that a sequence of wave heights can be treated as a Markov chain which allows for correlations between successive waves. Signicant improvements have been made since (e.g. Battjes and Van Vledder, 1984) and the method does a good job predicting run lengths in deep water. However, the run length itself is basically a measurement of groups in one dimension. It does not give an explicit description of the amplitude of the wave groups. 3. Groupiness Factor As opposed to the run length, the Groupiness Factor is a parameter that measures the amplitude modulation of the incident wave eld. As originally dened by Funke and Mansard (1979), the GF in terms of the SIWEH function E(t), is as follows: p m GF SIWEH = (1) m where m and m are the zeroth moment of the SIWEH spectrum S E (f), and the spectrum of the incident waves S (f), respectively. In this way, the GF relates the variance of the wave energy to the variance of the underlying process. Hudspeth and Medina (1988) point out that, for narrow-banded linear waves, if the dierence frequencies are exactly isolated in the energy time series, then S E (f) can be approximated by the following: S E (f) m ; (f) (11) 5
[E(t)] m (1) where ; (f) istheenvelope spectral density function as is dened by: ; (f) = m Z 1 S (x + f)s (x)dx: (13) It can be easily shown that the GF computed from Eq. 1 will be approximately unity. Any deviations from unity can be attributed to the ltering operation applied to E(t) or to deviations from linearity. List (199) suggests a similar GF dened in terms of an amplitude time series as: p A GF = (14) A(t) where A and A(t) are the standard deviation and the mean of the amplitude function A(t), respectively. Again, for narrow-banded linear waves the spectrum of A(t) can be approximated by S A (f) ( ; )m ; (f) (15) mean(a) 1 p m : (16) In a similar fashion, the GF determined from Eq. 14 will reduce to a constant ( :739). For these reasons the GF has limited ability to describe wave grouping. 3.3 Group Steepness In the following we dene a new parameter, group steepness, for quantifying wave grouping. Group steepness combines the amplitude of wave modulation and the number of waves per group in a matter similar to the ka parameter, which is a measure of wave steepness. The parameter as dened herein is not a predictive parameter, but rather a deterministic quantity which can be applied to wave data. The method of computing the group steepness is as follows: 1. remove low frequencies from wave time series ( :5Hz). compute A(t) with Hilbert transform (Eq. 8) 3. select wave group frequency band of interest (e.g. f :3Hz) 4. compute spectrum of S A (f) 5. compute group steepness (S x or S t ) 6
(cm) The group steepness can be dened spatially by the following: S x = A L g (17) where A is the standard deviation of A(t) dened spectrally as p m A. L g is the mean group wavelength as dened by small-amplitude wave theory, L g =(nl c T g )=T c (18) where n = C g =C is the ratio of phase speeds, T g =(m A =m A ) 1= is a spectral estimate of the mean group period, and T c =(m =m ) 1= is a spectral estimate of the mean wave period. L c is the mean wavelength found from T c and the dispersion relationship. The spatial steepness parameter can be applied to remotely sensed data where the data has a known spatial distribution. The temporal Group Steepness, S t, is dened as: S t = A : (19) gtg This dynamic parameter is a measure of the amplitude of the wave groups and their mean period. A wide-banded sea with essentially at groups will have large mean group period and hence small S t.moreover, this parameter can evaluate narrow-banded seas with similar mean group periods by distinguishing between high-amplitude groups and low-amplitude groups. It is expected the parameter can be applied to moored structures and will applied here to analyze the forcing of surf zone current modulations. a) 4 b) 4 4 6 S t =3:63 1 ;5 S x =1:8 1 ;3 GF =:939 5 1 15 5 3 35 time (sec) 4 6 S t =4:89 1 ;5 S x =5:5 1 ;3 GF =:9539 5 1 15 5 3 35 time (sec) Figure 1: Grouping parameters for synthetic waves in a) m b) m depth. To test the viability of the group steepness parameter, Figure 1a and b compares the group steepness and GF in simulated waves. Figure 1a shows a well modulated synthetic wave train in deep water. If the waves shoal according to linear theory their height will increase and visually their grouping will become 7
(cm) more pronounced. In Figure 1b the same synthetic waves are shown in two meters depth and S t has increased by 35% while the Groupiness Factor is essentially unchanged ( %). Since the wavelength of the waves has also shortened considerably, the spatial steepness has increased the most ( 3%). 4 Field Data Analysis Two eld data sets were analyzed. The rst set was obtained during the SU- PERDUCK experiments conducted by the U.S. Army Corps of Engineers in October of 1986 at the Field Research Facility at Duck, N.C. (Crowson et al., 1988). The beach prole was characterized by a linear bar system with the innermost bar located approximately 6m oshore. Pressure data was recorded by a cross-shore array of bottom-mounted pressure sensors located in depths ranging from 8 to 1m of water. Pressure data was converted to surface displacement using linear theory. In addition, current records were obtained by an alongshore array of Marsh-McBirney bidirectional electromagnetic current meters located approximately 5m from the shoreline in the trough of the innermost bar (Thorton and Guza, 1986). The wave elds considered here generally consist of long period swell from the south along with high-energy wind-generated waves from the north. Data for all sensors were sampled at Hz during 4-hour measuring periods centered about high and low tides. a) 5 b) 5 1 3 4 5 6 7 8 9 5 5 1 3 4 5 6 7 8 9 5 5 1 3 4 5 6 7 8 9 time (sec).5 1.5 1.5 (*) - S t (+) - S x (o) - GF 5 1 15 5 3 35 4 oshore dist. (m) Figure : a) Wave records at cross-shore position 365m (top), 11m (middle), and 7m (bottom) b) Grouping parameters vs. cross-shore distance, parameters are normalized by their oshore value -4-8 NSTS, Santa Barbara. Signicant wave breaking occured at 9m oshore The second data set was obtained during the Nearshore Sediment Transport Study conducted at Leadbetter Beach, Santa Barbara, California (198). The site consists of near planar beach topography (Thorton and Guza, 1986) its orientation is such to encourage incident waves with high angles of incidence. 8
(cm) (cm) u(cm/sec) u (cm/sec) Wave records were obtained from a cross-shore array of pressure sensors (1-9m) in a similar fashion to the previous data set. Figures a and b demonstrate the change in grouping parameters with shoaling for real wave data. The shoaled waves at x=11m ( a (middle)) are mostly unbroken and their grouping is more pronounced than at x=365m ( a (top)). Correspondingly, S t has increased 45%. Since the wavelegths of the waves have shortened during shoaling, while the numberof waves per groups has not changed signicantly, S x has increased to 5% of its oshore value. Groupiness Factor has increased 1 ; %. At the most inshore position, x =7m ( a (bottom)), waves are mostly broken and appear poorly grouped. Here S t is 75% of its oshore value. S x remains twice its oshore value, and GF is essentially unchanged from oshore. 1 a) 1 b) 5 5 5 5 1 1 15 5 6 7 8 9 1 11 1 13 14 15 time (min) 15 1 3 4 5 6 7 8 9 1 time (min) Figure 3: Cross-shore velocity records at a)t=.1hr b)t=3.33hr current meter LX, 1-1-86 SUPERDUCK. 5 a) 4 3 1 1 3 4 b) 15 1 5 5 1 15 5 1 1.5 11 11.5 1 1.5 13 13.5 14 14.5 15 time (min) 199 199.5.5 1 1.5.5 3 3.5 4 time (min) Figure 4: Wave record at a)t=.hr b)t=3.33hr pressure gage LA6, 1-1-86 SUPERDUCK. Note scale change. Figures 3 and 4 show the cross-shore current record measured in the inner surf zone and the oshore wave record at two dierent times during data collection on October 1, 1986 at SUPERDUCK. The current record indicates a spin-up 9
of a long-period (O(1s)) current oscillation that can be termed a shear wave or large scale eddy in the nearshore circulation (Oltman-Shay et al., 1989). The wave records in Figure 4a and b demonstrate an increase in waveheight and wave grouping as a storm blew in from the northeast. Figure 5 compares the variation of grouping parameters with time. The values of GF showed signicant net increase from t=1hr to t=3.3hr only at the oshore gage, while the GF at all the surfzone gages showed a net decrease. Both group steepness parameters increased at all gages, with the oshore gage showing the largest increase. It is suggested that the signicant increase of group steepness in the surfzone over time, which corresponds to the growing long-period modulations of the wave eld, indicates that wave groups play a role in the inducement of long-period current oscillations, or shear waves, measured in the surfzone. GF 8 S x 8 S t x=914m x=914m 1.8 7 7 1.6 6 6 17m 1.4 5 5 x=914m 1. m 4 4 1 187m 3 17m 3 17m.8 187m.6.4 1 3 1 m 1 3 187m 1 m 1 3 time (hr) Figure 5: Grouping parameters vs. time. All parameters are normalized by the value at oshore gage LA6 at t=1hr. Distance of sensor from origin is also given 1-1-86 SUPERDUCK. 5 Conclusions Two parameters are introduced to quantify wave grouping by utilizing information regarding modulation of the incidentwave eld and its spatial and temporal scales. The parameters are shown to be more responsive than GF to changes in the wave eld due to shoaling and breaking. The parameters are based on using 1
the Hilbert transform to determine the wave envelope from the wave record. This method is shown to be superior to other methods involving and jj in both accuracy and speed of computation. Analysis of eld data demonstrates that signicant wave grouping remains after waves have broken. This gives further evidence that models of long wave generation should account for nonsteady radiation stress gradients in the surf zone. In addition, increased group steepness is shown to correlate well with increased long period modulation of nearshore currents. Acknowledgements. Funding for this study was provided by the Army Research Oce under project DAAL3-9-G-116. The authors would also like to thank Joan Oltman-Shay for her help in obtaining SUPERDUCK and NSTS data sets. 6 References Battjes, J.A., and Vledder, V. (1984). \Verication of Kimura's theory for wave group statistics," Proceedings 19th ICCE, Houston, ASCE, 1, 64-648. Bowen, Anthony J., and Guza, R.T. (1978). \Edge waves and surf beat," Jour. of Geophys. Res., 83, No. C4, 1913{19. Bowers, E.C., (1988). \Wave grouping and harbor design," Coastal Engineering, June, 37-58. Crowson, Ronald A., Birkemeier, William A., Klein, Harriet M., and Miller, Herman C. (1988). \SUPERDUCK nearshore processes experiment: Summary of studies. CERC Field Research Facility," Technical Report CERC-88-1, US Army Engineer Waterways Experiment Station, Coastal Engineering Research Center, Vicksburg, MS. Dean, R.G. (198). \Kinematics and forces due to wave groups and associated second-order currents," Proceedings Second International Conference onthebehavior of Oshore Structures, (BOSS'8), London, paper No.6. Elgar, S., Guza, R.T., and Seymour, R.J. (1984). \Groups of waves in shallow water," Jour. of Geophys. Res., 89, No. C3, 363-3634. Funke, E.R., and Mansard, E.P.D. (1979). \Synthesis of realistic sea states in a laboratory ume," Hydraul. Lab. Tech. Rep. LTR-HY-66, National Research Council of Canada. Goda, Y. (197). \Numerical experiments on wave statistics with spectral simulation," Rep. Port Harbour Res. Inst., Nagase, Yokosuka, Japan, 8, 3-57. Hamilton, R.P., Dalrymple, R.A., Oltman-Shay, J., Putrevu, U., and Haller M.C., \Wave group forced nearshore circulation," unpublished. 11
Hsu, F.H., and Blenkarn, K.A. (197). \Analysis of peak mooring forces caused by slow vessel drift oscillations in random seas," Proceedings OTC Conference, Houston, TX, Paper No. 1159. Hudspeth, Robert T., and Medina, Joseph R. (1988). \Wave group analysis by Hilbert transform," Coastal Engineering, Vol. 1, 885-898. List, Jerey H. (199). \Wave groupiness in the nearshore," Coastal Engineering, Vol. 15, 475-496. Longuet-Higgins, M.S., and Stewart, R.W. (196). \Radiation stress in water waves: a physical discussion, with applications." Deep Sea Research, 11, 59-56. Medina, J.R., and Hudspeth, R.T. (1987). \Sea states dened by wave height and period functions," IAHR Seminar on Wave Analysis and Generation in Laboratory Basins, nd IAHR Congress, Lausanne, Switzerland, 49-59. Melville, W.K. \Wave modulation and breakdown", J. Fluid Mech., vol. 18, 489-56. Naess, Arvid (1978). \On experimental prediction of low-frequency oscillations of moored oshore structures," Norwegian Marine Research, No.3, 3-37. Oltman{Shay, Joan, P.A. Howd, and W.A. Birkemeier (1989). \Shear instabilities of the mean longshore current,. eld observations," J. Geophys. Res., 94, 1831{184. Pinkster, J.A. (1975). \Low frequency phenomena associated with vessels moored at sea," Jour. Soc. Pet. Eng., May:474-494. Rice, S.O. (1944). \Mathematical analysis of random noise," BellSystemTechnical Journal, 3, 8-33. Rice, S.O. (1945). \Mathematical analysis of random noise," BellSystemTechnical Journal, 4, 46-156. Symonds, G., Huntley, D.A., and Bowen, A.J. (198). \Two-dimensional surf beat: long wave generation by a time-varying breakpoint," J. Geophys. Res., 87, 49-498. Tang, Ernest C.-S., and Dalrymple, R.A. (1989). \Nearshore circulation: rip Currents and wave groups," in Nearshore Sediment Transport Study, R.J. Seymour ed., Plenum Press, NY. Thompson, E.F., and Seelig, W.N. (1984). \High wave grouping in shallow water," Jour. of Waterway, Port, Coastal, and Ocean Engineering, 11 (),139-157. Thorton, E.B., and R.T. Guza (1986). \Surf zone alongshore currents and random waves: Field data models," Jour. Phys. Oceanogr., 16,1165-1178. Tucker, M.J. (195). \Surf beats: sea waves of 1 to 5 min. period," Proceedings of the Royal Society, A, Vol., 565-573. 1